In this paper, we define the vanishing-off subgroup of a nonabelian group. We study the structure of the quotient of this subgroup and a central series obtained from this subgroup. MSC primary: 20C15, MSC secondary: 20D25

In this course we will be studying the subgroup structure of the finite alternating and symmetric groups. What does the phrase “study the subgroups of symmetric groups” mean? In this introduction I’ll suggest an answer to that question, and attempt to convince you that answer has some merit. In the process you’ll get some idea of the material we will be covering, and I’ll attempt to motivate th...

a group $g$ is said to be a c-tidy group if for every element $x in g setminus k(g)$‎, ‎the set $cyc(x)=lbrace y in g mid langle x‎, ‎y rangle ; {rm is ; cyclic} rbrace$ is a cyclic subgroup of $g$‎, ‎where $k(g)=underset{x in g}bigcap cyc(x)$‎. ‎in this short note we determine the structure of finite c-tidy groups‎.

let $h$, $l$ and $x$ be subgroups of a finite group$g$. then $h$ is said to be $x$-permutable with $l$ if for some$xin x$ we have $al^{x}=l^{x}a$. we say that $h$ is emph{$x$-quasipermutable } (emph{$x_{s}$-quasipermutable}, respectively) in $g$ provided $g$ has a subgroup$b$ such that $g=n_{g}(h)b$ and $h$ $x$-permutes with $b$ and with all subgroups (with all sylowsubgroups, respectively) $v$...